That is correct. I don't believe the Roman Catholic Church has ever taken an official position on the exact day and, further, the calendar changes which occurred subsequently could throw it off by two weeks or more which is, I believe, why the Orthodox Church celebrates it on January 6th.
Cromwell and his puritan allies did try to ban the celebration of Christmas with limited success in the 1650s because they bought into the 'pagan holiday' theory.
December 25th is not that far from the winter solstice and could very well be the first day that ancient pagans could actually tell the days were starting to get longer, so I do not believe Cromwell & crew's theory was completely without basis.
Whatever the motivation, the symbolism that Jesus came to earth at its darkest time and begin to lengthen the days is more important to almost all sects of Christendom than the actual day.
And the corollary to said symbolism is that he will return to earth again in its darkest days, the days which the odd alliance of godless humanists, liberals and Islamofacists are putting us on a course for now.
Unfortunately the actual length of the year is not exactly 365.25 days. Every 128 years the calendar was off by one day, which is why Pope Gregory had to reform the calendar in 1582 by dropping 10 days (so the day after October 4 was October 15). He wanted to get the calendar back to what it was in A.D. 325 at the time of the Council of Nicaea, when the rules for calculating Easter were adopted. At that time the vernal equinox was March 21. The winter solstice falls on December 21 or 22. The reason it doesn't fall on December 25 is because Pope Gregory wasn't interested in getting the calendar back to how it was in Julius Caesar's day.
Actually the Orthodox Church celebrates the Feast of the Nativity on December 25th.
Those on the Old (Julian) Calendar — Russia, Serbia, Georgia, Jerusalem — just happen to have the Julian December 25th coincide with the civil and Gregorian calendars’ January 7th — for now — the drift caused by the Gregorian calendar skipping leap years at the beginning of some centuries will push it to January 8th eventually.
Those on the New “Reformed Julian” Calendar (which coincided with the Gregorian until, should the Lord tarry, the 26th century when it will diverge, becoming more astronomically accurate, again absent intervening calendar reforms) celebrate on December 25th on that calendar, which coincides with December 25th on the civil and Gregorian calendars.